Changes in potentials due to changes in the point spectrum: Anharmonic oscillators with exact solutions

Assume a solved quantum-mechanical problem for the one-dimensional Schr\"odinger equation, which has a discrete spectrum. An algorithm is presented to calculate exactly the change in the original potential and eigenfunctions brought about by arbitrary changes in the positions of the original eigenvalues and/or the normalizations of the corresponding eigenfunctions. As a first example, we consider the modification of the harmonic oscillator and thereby obtain potentials for anharmonic oscillators with exact eigenvalues and eigenfunctions. Next, we introduce potentials in the one-dimensional box with rigid walls which alter the usual spectrum in a predetermined way. We also sketch the process of obtaining a change in the Coulomb potential which deletes the lowest eigenvalue for the zero-angular-momentum equation.